Four letters to different insureds are prepared along with accompanying envelopes. The letters are put into the envelopes randomly.

Calculate the probability that at least one letter ends up in its accompanying envelope.

• 27/256
• 1/4
• 11/24
• 5/8
• 3/4

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: D

Consider the two mutually exclusive events “first envelope correct” and “first envelope incorrect.” The probability of the first event is 1/4 and meets the requirement of at least one correct. For the 3/4 of the time the first envelope is incorrect, there are now 3 more envelopes to fill. Of the six permutations, three will place one letter correctly. The total probability is 1/4 + 3/4(3/6) = 5/8.